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## Homework Statement

i) Solve the equation [itex] z^3 = \mathbf{i}[/itex].

**(ii)**

**Hence find the possible values for the argument of a complex number w which is such that**[itex] w^3 = \mathbf{i}(w*)^3[/itex].

I'm stuck on part ii.

## Homework Equations

## The Attempt at a Solution

The answer to the equation in part i: e^(1/6 pi i) , e^(5/6pi i ), e^(-1/2pi i )

In the back of the book it has 6 answers, I can only get three of them. If I draw it out on the argand diagram the 3 answers I am missing would be the arguments if you carried the line on through the axis. For example I have arguments 5/12pi , pi/12, -pi/4 and am missing 3/4pi, -11/12pi and -7/12pi.

My method was to cube root both sides:

e^theta i = Z e^ -theta i

Where Z is one of the solutions to part i, by changing Z I can get 3 solutions but am missing the other three.

After seeing in the book there were six solutions I did manage to get them by letting i = e^pi/2 i and setting it equal to e^6theta i, then I just kept adding 2pi to the argument of i until I got all 6 arguments and ended up back at the beginning. However if I didn't have the answers I wouldn't have known to look for the extra three, and this method didn't involve part i) of the question (which I think it is meant to because of the 'hence').

I'd be interested to see how you all approach the problem, and an explanation of what is actually happening.

Thanks